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Renewal theory in $r$ dimensions (I)

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Renewal theory in r dimensions (I) COMPOSITIO MATHEMATICA A. J. STAM Renewal theory in r dimensions (I) Compositio Mathematica, tome 21, no 4 (1969), p. 383-399. <http://www.numdam.org/item?id=CM_1969__21_4_383_0> © Foundation Compositio Mathematica, 1969, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 383 Renewal theory in r dimensions (I) by A. J. Stam COMPOSITIO MATHEMATICA, Vol. 21, Fasc. 4, 1969, pag. 383-399 Wolters-Noordhoff Publishing Printed in the Netherlands Summary Let X1, g 2’ ... be strictly d-dimensional independent random vectors with common distribution function F, with finite second moments and nonzero first moment vector fi. Let U(A ) = 03A3~1 Fm(A), where Fm denotes the m-fold convolution of F. The paper studies the asymptotic behaviour as Ixl - oo of U(A +x) for bounded A. The results of Doney (Proc. London Math. Soc. 26 (1966), 669-684) are derived under more general conditions by a new technique, viz. by first studying the more easily manageable generalized renewal measure WF = 03A3~1 m03C1 Fm, where p = 1 2(d-1). This is done by comparing WF and WG for F and G having the same first and second moments, using local central limit theorems. 1. Introduction Throughout this paper F, G and H will denote distribution functions of strictly d-dimensional probability measures - also denoted by F, G, H - with characteristic functions ~, y, X, respectively. A measure on the Borelsets of Rd is called strictly d-dimensional if its support is not contained in a hyperplane of dimensi

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